#205104

order	512
nilpotency class	3
isoclinism	isoclinic class of #19456
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#390
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

955 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,13)(4,16)(5,10)(6,11)(7,12)(8,14)(9,15)(18,22,24,27)(23,29,30,32);(1,3,5,14,7,15,6,4)(2,8,11,13,12,16,10,9)(17,18,20,27,21,24,28,22)(19,23,26,32,25,30,31,29);(1,4,6,15,7,14,5,3)(2,9,10,16,12,13,11,8)(17,19)(18,23)(20,26)(21,25)(22,29)(24,30)(27,32)(28,31);(1,5,7,6)(2,10,12,11)(3,4,15,14)(8,13,16,9)(19,25)(23,30)(26,31)(29,32)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,d^2=h,e^2=hi,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,e^b=ei,g^b=gi,d^c=dhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112300);